A lithographic apparatus is a machine that applies a desired pattern onto a substrate, usually onto a target portion of the substrate. A lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). In such a case, a patterning device, which is alternatively referred to as a mask or a reticle, may be used to generate a circuit pattern to be formed on an individual layer of the IC. This pattern can be transferred onto a target portion (e.g. including part of, one, or several dies) on a substrate (e.g. a silicon wafer). Transfer of the pattern is typically via imaging onto a layer of radiation-sensitive material (resist) provided on the substrate. In general, a single substrate will contain a network of adjacent target portions that are successively patterned. Conventional lithographic apparatus include so-called steppers, in which each target portion is irradiated by exposing an entire pattern onto the target portion at once, and so-called scanners, in which each target portion is irradiated by scanning the pattern through a radiation beam in a given direction (the “scanning”-direction) while synchronously scanning the substrate parallel or anti-parallel to this direction. It is also possible to transfer the pattern from the patterning device to the substrate by imprinting the pattern onto the substrate.
A lithographic apparatus generally includes moveable objects, such as for instance a support constructed to support the patterning device or a substrate table to hold the substrate. In these cases, positioning systems are provided to position the moveable objects with respect to another object, such as a frame, for instance a metrology frame or a base frame. Positioning systems generally include an actuator to apply a force to the moveable object in dependency of a drive signal provided by a control system.
The control system is configured to control a position quantity of the moveable object, such as a position, velocity, acceleration, jerk, etc., and can be an absolute or relative position quantity. In high-accuracy applications such as a lithographic apparatus, it is desirable that the positioning system provides closed-loop control and therefore includes a sensor to measure the actual position quantity of the moveable object, which is “fed back” into the control system as an input. Open-loop control without the use of a sensor may suffice if the required accuracy is relatively low.
Generally, a control system includes a set-point generator to provide a reference signal that represents a desired position quantity of the moveable object. The control system further includes a subtractor to compare the actual position quantity with the desired position quantity. The difference between the signals representing the actual and desired position quantity, i.e. the reference sensor and the signals provided by the sensor, is supplied to a control unit, which drives the actuator based on the difference (also known as error signal). The control unit usually includes a feedback controller and may include a feed-forward controller that provides an additional drive signal in dependency of a reference signal only.
The aim when designing a control system is to improve the low-frequency disturbance suppression while maintaining favorable high-frequency noise properties. However, bound by inherent design limitations a linear feedback control system generally fails in satisfying this aim to the fullest, especially when the behavior of the combination of actuator and moveable object is varying with the position. In that case, it may be difficult to optimize the control system over the entire positioning range. The varying behavior can be due to motor position dependent actuator properties (e.g. a motor position dependent motor constant), or when properties of parts of the moveable object are position dependent (e.g. a spring constant that is position dependent). To exceed beyond the possibilities given by a nominal (and linear) control system, nonlinear control can be used.
An example of nonlinear control is a variable gain control, wherein the gain of the control system is not constant for a given frequency, but depending on, for instance, the difference between the actual and desired position quantity of the moveable object. This increases the flexibility in dealing with position-dependent disturbances and thereby improves the low-frequency disturbance suppression. However, a drawback of this approach is that at the same time the high-frequency noise properties may deteriorate. This reduces the obtainable position accuracy of the positioning system and may result in imaging problems and/or overlay errors in the lithographic apparatus.